Large sparse circuits of spiking neurons exhibit a balanced state of highly irregular activity under a wide range of conditions. It occurs likewise in sparsely connected random networks that receive excitatory external inputs and recurrent inhibition as well as in networks with mixed recurrent inhibition and excitation. Here we analytically investigate this irregular dynamics in finite networks keeping track of all individual spike times and the identities of individual neurons. For delayed, purely inhibitory interactions we show that the irregular dynamics is not chaotic but in fact stable. Moreover, we demonstrate that after long transients the dynamics converges towards periodic orbits and that every generic periodic orbit of these dynamical systems is stable. We investigate the collective irregular dynamics upon increasing the time scale of synaptic responses and upon iteratively replacing inhibitory by excitatory interactions. Whereas for small and moderate time scales as well as for few excitatory interactions, the dynamics stays stable, there is a smooth transition to chaos if the synaptic response becomes sufficiently slow (even in purely inhibitory networks) or the number of excitatory interactions becomes too large. These results indicate that chaotic and stable dynamics are equally capable of generating the irregular neuronal activity. More generally, chaos apparently is not essential for generating high irregularity of balanced activity, and we suggest that a mechanism different from chaos and stochasticity significantly contributes to irregular activity in cortical circuits.